When a wallet is scanning for incoming transactions, every transaction is scanned to see if it is for "you". This only requires your private view key and your public spend key, and this check is immutable and cannot be faked. You cannot receive transactions and identify them without a corresponding private view key.
In order to spend the funds you have to compute a one-time private spend key for that output.This is almost always done automatically by the Monero Wallet software.